7,089 research outputs found
An Exact Algorithm for TSP in Degree-3 Graphs via Circuit Procedure and Amortization on Connectivity Structure
The paper presents an O^*(1.2312^n)-time and polynomial-space algorithm for
the traveling salesman problem in an n-vertex graph with maximum degree 3. This
improves the previous time bounds of O^*(1.251^n) by Iwama and Nakashima and
O^*(1.260^n) by Eppstein. Our algorithm is a simple branch-and-search
algorithm. The only branch rule is designed on a cut-circuit structure of a
graph induced by unprocessed edges. To improve a time bound by a simple
analysis on measure and conquer, we introduce an amortization scheme over the
cut-circuit structure by defining the measure of an instance to be the sum of
not only weights of vertices but also weights of connected components of the
induced graph.Comment: 24 pages and 4 figure
Constant-factor approximations for Capacitated Arc Routing without triangle inequality
Given an undirected graph with edge costs and edge demands, the Capacitated
Arc Routing problem (CARP) asks for minimum-cost routes for equal-capacity
vehicles so as to satisfy all demands. Constant-factor polynomial-time
approximation algorithms were proposed for CARP with triangle inequality, while
CARP was claimed to be NP-hard to approximate within any constant factor in
general. Correcting this claim, we show that any factor {\alpha} approximation
for CARP with triangle inequality yields a factor {\alpha} approximation for
the general CARP
The Symmetric Group Defies Strong Fourier Sampling
The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states
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